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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Lie algebra treatment of three coupled oscillators

M. Sebawe Abdalla, M. A. Bashir

Published Apr 1, 1998• 13 citations
Publisher

Abstract

Using the Lie algebra technique we approach the problem of three-mode interaction (Stokes, anti-Stokes and phonon). The relationship between the Hamiltonian model and the directional coupler is considered. The most general solution of the wavefunction (Schrödinger representation) is obtained. The advantage of this technique compared with the classical method is given.

DOI
10.1088/1355-5111/10/2/013

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Using the Lie algebra technique we approach the problem of three-mode interaction (Stokes, anti-Stokes and phonon). The relationship between the Hamiltonian model and the directional coupler is considered. The most general solution of the wavefunction (Schrödinger representation) is obtained. The advantage of this technique compared with the classical method is given.
We propose a solution to the problem of solving the Schr\"odinger equation for an arbitrary number of identical one-dimensional harmonically coupled oscillators raised by Fan Hong-yi [Phys. Rev. A 42, 4377 (1990)]. The relationship between the Fock spaces associated with the uncoupled and coupled oscillators is given as well as the coordinate representation of the eigenstates. In view of further applications, the Lie algebraic properties of the model are examined, and the generalization to three spatial dimensions is made.
This work is aiming to show the advantage of using the Lie algebraic decomposition technique to solvefor Schrödinger’s wave equation for a quantum model, compared with the direct method of solution. The advantageis a two-fold: one is to derive general form of solution, and, two is relatively manageable to deal with the case oftime-dependent system Hamiltonian. Specifically, we consider the model of 2-level optical atom and solve for thecorresponding Schrödinger’s wave equation using the Lie algebraic decomposition technique. The obtained formof solution for the wave function is used to examine computationally the atomic localization in the coordinate space.For comparison, the direct method of solution of the wave function is analysed in order to show its complicationwhen dealing with time-dependent Hamiltonian.The possibility of using the Lie algebraic method for a qubit model(a driven quantum dot model) is briery discussed, if Schrödinger’s wave function is to be examined for the qubitlocalization.
In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix.The quantization of a Hamiltonian describing such a system has been done in the canonical case.In this paper, we take a more general approach and look at the system as a Wigner quantum system.Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations.Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n).We determine the spectrum of the considered Hamiltonian in specific representations of these Lie superalgebras and discuss the results in detail.We also make the connection with the well-known canonical case.
We find that a three-mode boson realization of the SU(1,1) Lie algebra is involved in solving the three coupled harmonic oscillators' problem. The unitary operator U, which is found to be able to transform the Fock space of a three–dimensional isotropic harmonic oscillator into the space in which the Hamiltonian of three coupled oscillators is diagonized, is further decomposed as a quantum–mechanical rotation in three–mode Hilbert space followed by an SU(1,1) squeezing transformation. The coordinate representation of this SU(1,1) unitary operator is obtained.